Distributed consensus is an important special case of Problem original with \(f_i(\mathbf {x})=\Vert \mathbf {x}-\mathbf {x}_i\Vert ^2,\forall i\in \mathcal {V}\) , where nodes are supposed to converge to the average \(\bar{\mathbf {x}}=\frac{1}{n}\sum _{i=1}^n\mathbf {x}_i\) . A celebrated algorithm to solve it is proposed in [1]} with the update rule \(\mathbf {x}_i^{k+1}= \sum _{j\in \mathcal {N}_i}[W]_{ij}\mathbf {x}_j^k, \forall i\in \mathcal {V}\)  and \(\mathbf {x}_i^0=\mathbf {x}_i\) . Let \(X^k=[\mathbf {x}_1^k,\cdots ,\mathbf {x}_n^k]^\mathsf {T}\) . Its compact form is given as
\(X^{k+1}=WX^k.\) 
